1. Field of the Invention
The present invention relates to a pattern recognition method adapted to approximate the distribution of a set of vectors and the boundary of two or more sets (classes) of vectors in a vector space based on basis functions.
2. Description of the Related Art
Methods of using basis functions referred to as radial basis functions (to be referred to as spherical basis functions hereinafter) are known. Spherical basis functions have been proposed by several study groups independently. J. E. Moody and C. Darken: “Fast Learning in Networks of Locally-Tuned Processing Units”, Neural Computation 1, pp. 281-294, 1989, may be cited here as an example of such propositions. Spherical basis functions have a peak at the center and symmetrical in all directions. Of spherical basis functions, those of the so-called Gaussian type are the most popular and are expressed by
            o      i        ⁡          (      x      )        =      exp    ⁡          [                                    -                                                                            x                  -                                      ξ                    i                                                                              2                                /          2                ⁢                  σ          i          2                    ]      where x is the vector that corresponds to the input pattern and ξi is the i-th basis vector (parameter indicating the position in the Gaussian distribution) while σi is the i-th standard deviation (parameter indicating the expanse of the Gaussian distribution). The value of the i-th Gaussian type basis function is oi(x), which is not negative and large when x is close to ξi and takes the largest value of 1 when x=ξi. It is possible to approximate the distribution of any arbitrarily selected set of vectors to a desired accuracy level by providing a sufficient number of basis functions and using a weighted linear combination as expressed by
            y      1        ⁡          (      x      )        =            ∑              i        =        1                    H        ′              ⁢                  w                  1          ⁢          i                    ⁢                        o          i                ⁡                  (          x          )                    where l is the class number of the set of vector and wli is the contribution ratio (weight parameter) of the i-th basis function to the class l, while H′ is the number of basis functions. The above formula indicates the extent to which an unknown input pattern resembles the particular class (degree of similarity) so that it can be used to classify classes. For example, ifC(x)=arg max1[y1(x)]it is possible to determine the class of an input pattern according to the class boundary defined by basis functions. In the formula 3 above, argmaxl [·] is the number of the class that provides the largest value for the degree of similarity.
The pattern recognition method that uses spherical basis functions provides advantages including the ability to optimize parameters by learning, like feedforward neural nets based on general sigmoid functions; but, unlike general neural nets, the contribution ratios of individual basis functions are intuitively comprehensible.
However, the distribution of vectors that corresponds to a pattern observed from the real world is, more often than not, complex and hence it is necessary to prepare a large number of basis functions in order to accurately approximate such a distribution. Conversely, when the number of obtained samples is small, approximation can produce a state of being too complex relative to the proper distribution (population distribution) (excessive learning).